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Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio

Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio. Robert J Poole Department of Engineering, University of Liverpool, UK. Manuel A Alves CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal. Paulo J Oliveira

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Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio

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  1. Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio Robert J Poole Department of Engineering, University of Liverpool, UK Manuel A Alves CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal Paulo J Oliveira Departamento de Engenharia Electromecânica, Universidade da Beira Interior, Portugal Fernando T Pinho aCEFT, Faculdade de Engenharia, Universidade do Porto, Portugal bUniversidade do Minho, Portugal AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  2. Outline • Introduction • Governing equations • Numerical method / grid dependency issues • Newtonian results • UCM simulations: “High” ER followed by “Low” ER • Conclusions AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  3. Introduction Why investigate expansion flows of viscoelastic liquids? Prevailing view….vortex suppressed by elasticity and totally eliminated at “high” Deborah Not the whole story (AERC 2006 Poole et al, JNNFM 2007 to appear) UCM/Oldroyd-B (β= 1/9) simulations, 1:3 expansion ratio, creeping flow • Maximum obtainable De ≈ 1 • Effect of elasticity is to reduce but not eliminate recirculation • Enhanced pressure drop observed AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  4. Governing equations 1) Mass 2) Momentum (creeping flow) 3) Constitutive equation Upper Convected Maxwell model (UCM) • Essentially phenomenological model • “Simplest” viscoelastic differential model • Capable of capturing qualitative features of many highly-elastic flows AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  5. Numerical method 1) Finite-volume method (Oliveira et al (1998), Oliveira & Pinho (1999)) 2) Structured, collocated and non-orthogonal meshes 3) Discretization (formally second order) Diffusive terms: central differences (CDS) Convective terms: CUBISTA (Alves et al (2003)) 4) Special formulations for cell-face velocities and stresses AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  6. L1= 20d L2= 100d ER=D/d h D d UB X symmetry axis Y Computational domain and meshes Neumann b.c.s at exit Expansion ratios (ER) 1:1.5 1:2 Low ER 1:3 1:4 1:8 High ER 1:16 Fully-developed inlet velocity and stress profiles 1:32 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  7. Representative mesh details AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  8. Representative grid dependency and numerical accuracy #denotes extrapolated value using Richardson’stechnique AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  9. Newtonian simulations: XR variation with ER d Linear fit to data for ER  4 (R2=1) AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  10. Newtonian simulations: XR variation with ER Deviations from linear fit as ER  1 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  11. Newtonian simulations: XR variation with ER H D AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  12. “High” ER viscoelastic : XR variation with De and ER Δ M1 X M2  Extrapolated AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  13. 1:4 expansion ratio AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  14. 1:4 expansion ratio (M2) De = 0.0 De = 0.2 De = 0.4 De = 0.6 De = 0.8 De = 1.0 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  15. “High” ER viscoelastic : scaling of XR ER =32 ER =16 ER =8 ER =4 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  16. “Low” ER viscoelastic : XR variation with De and ER AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  17. 1:1.5 expansion ratio 1:2 expansion ratio AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  18. 1:1.5 expansion ratio De = 0.6 De = 0.4 De = 0.3 De = 0.2 De = 0.1 De = 0.0 De = 0.8 De = 1.0 De = 0.8 De = 0.6 De = 0.0 De = 0.1 De = 0.4 De = 0.3 De = 1.0 De = 0.2 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  19. “Low” ER viscoelastic : scaling of XR AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  20. Maximum De  1.0? McKinley et al scaling criterion for onset of purely elastic instabilities: independent of ER Streamlines at De = 1 for ER = 4, 8 and 16 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  21. Maximum De  1.0? McKinley scaling criterion for onset of purely elastic instabilities: Streamlines at De = 1 for ER = 4, 8 and 16 AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  22. Conclusions For large expansion ratios (  8) • In range of De for which steady solutions could be obtained XR • decreases with elasticity • Recirculation length normalised with downstream duct height scales with a Deborah number based on bulk velocity at inlet and downstream duct height (De/ER) For small expansion ratios (  2) • XR initially decreases before increasing at a given level of elasticity (De/ER~ 0.4) Maximum obtainable De is approximately 1.0: independent of ER AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  23. Enhanced pressure drop AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  24. ‘2D’ 1: 13.3 Planar Expansion Townsend and Walters (1993) Re < 10 De O(1)? Newtonian 0.15% polyacrylamide solution AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  25. Stress variation around sharp corner r Hinch (1993) JnNFM Stresses around sharp corner go to infinity as: AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

  26. Normal stresses (ER = 3) AERC 20074th Annual European Rheology Conference April 12-14, Napoli - Italy

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